## Hereditary rings containing an injective maximal left ideal.(English)Zbl 0803.16010

R. Yue Chi Ming has proposed the following questions in his earlier papers. (i) If a ring $$R$$ contains an injective maximal left ideal and if every maximal left ideal of $$R$$ is projective, then is $$R$$ semisimple Artinian? (ii) If $$R$$ is a left hereditary ring containing an injective maximal left ideal, is $$R$$ semisimple Artinian?
The authors answer both the questions in the negative by constructing a hereditary ring $$R$$ containing an injective maximal left ideal as well as an injective maximal right ideal such that $$R$$ is Artinian with nonzero Jacobson radical $$J$$ with $$J^ 2 = 0$$. They also give two new characterizations of semisimple Artinian rings in terms of left hereditary rings containing an injective maximal left ideal.

### MSC:

 16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. 16D25 Ideals in associative algebras 16P20 Artinian rings and modules (associative rings and algebras)
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### References:

 [1] Yue Chi Ming R., Arch Math [2] Yue Chi Ming R., Comm. in Algebra 20 (3) pp 749– (1992) · Zbl 0754.16007 [3] Yue Chi Ming R., Riv. Mat. Univ., Parma 8 (3) pp 443– (1982) [4] Yue Chi Ming R., Ricerche di Matematica (2) pp 147– (1984) [5] Osofsky B.L., J. Math 14 (2) pp 645– (1964) [6] Faith C., Algebra I: Rings, Modules and Categories (1981) · Zbl 0508.16001 [7] Rege M.B., Math. Japonica 31 (6) pp 927– (1986) [8] Anderson F.W., Rings and Categories of Modules (1974) · Zbl 0301.16001 [9] Googearl K.R., Rings Theory: Nonsingular Rings and Modules (1976)
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